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Introducing the Quadratic Function and Its Graph

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Quadratic Functions

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Introducing the Quadratic Function and Its Graph

Algebra-1

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You will explore the concept of quadratic functions studying the free fall of a ball and the jump of a skier.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Identify the difference between a linear and a quadratic function.
  • Graph quadratic functions by making a table of values.
  • Identify the vertex, domain, range, x-intercept, y-intercept and axis of symmetry of a quadratic function on its graph.

Everything You'll Have Covered

An expression or equation is called quadratic if it has a degree of two.

The highest exponent term in the equation or function should have a power of two or an x squared term. If the highest power is a one then we would have a linear equation.

When we look at a table of values for a quadratic function, the pattern is not a linear one where each of the terms follows the same pattern. In a quadratic function, we will see that the pattern is continuing to increase from term to term. In a table of values for a quadratic function, you need to look at the patterns in both the first and second differences. When the second differences follow the same pattern, then you have a quadratic function.

The General Form of a Quadratic Equation is f(x) = ax^2 + bx + c.

This is the general form of a quadratic equation. The a, b, and c are real numbers. Notice it has an x squared term, a linear term, and a constant. The 'a' term cannot be zero because then you would not have the x squared. Either of the other terms could be zero.

A quadratic function is one in which the resulting graph is a parabola.

The graph of a quadratic is going to be a parabola or a "U" shaped graph. It can be pointing upward or downward depending on the leading term or coefficient being positive or negative. A parabola is symmetrical in shape so the one side looks the same as the other side.

The following key vocabulary terms will be used throughout this Activity Object:

axis of symmetry- the line about which a figure is symmetrical (like a mirror).

domain- the set of input values for which a relation/function is defined.

function - a special type of relation in which each element of the domain is paired with exactly one element of the range.

linear relationship - a relationship where the terms are constant and the resulted graph is a line.; it is in the form f(x) = mx +b.

maximum - the highest point on a graph (vertex if the parabola is downward).

minimum - the lowest point on a graph (vertex if the parabola is upward).

parabola - the U-shaped graph of a quadratic function f(x) = ax2 + bx + c, where a ? 0.

quadratic function - a nonlinear function written in general form f(x) = ax2 + bx + c, where a, b, c are real numbers and a ? 0.

range - set of all output values produced by a function.

vertex - the lowest point (minimum) on a parabola opening up or the highest point (maximum) on a parabola opening down; the point at which a parabola and its axis of symmetry intersect.

x-intercept - the x-coordinate of the point where the parabola intersects the x-axis. 

y-intercept - the y-coordinate of the point where the parabola intersects the y-axis.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Learners should be familiar with the concept of functions and linear functions.
Course Algebra-1
Type of Tutorial Concept Development
Key Vocabulary parabola, quadratic functions, graphing functions